On computational formulas for parametric type polynomials and its applications

Öz

In this paper, many formulas and identities for computing the r-parametric Hermite type polynomials are given with the help of generating functions. Using generating functions and algebraic methods, a relation is also given including these polynomials and the 2-variable Hermite Kampé de Fériet polynomials. Moreover, many relations and formulas containing the two parametric type of Apostol-Bernoulli polynomials of higher order, the two parametric type of Apostol-Euler polynomials of higher order, the two parametric type of Apostol-Genocchi polynomials of higher order and the Dickson polynomials are obtained. Finally, some special values of these polynomials and their applications with trigonometric functions are presented.

Anahtar Kelimeler

• Hermite type polynomials
• Apostol type numbers and polynomials
• parametric type polynomials
• special numbers and polynomials
• generating functions

Parametrik tip polinomlar için hesaplama formülleri ve uygulamaları

Öz

Bu çalışmada, r-parametreli Hermite tipli polinomların hesaplanması için birçok formüller ve bağıntılar, üreteç fonksiyonları yardımıyla verilmiştir. Üreteç fonksiyonları ve cebirsel yöntemler kullanılarak, bu polinomları ve 2-değişkenli Hermite Kampé de Fériet polinomlarını içeren bir bağıntı da verilmiştir. Ayrıca, yüksek mertebeden iki parametreli Apostol-Bernoulli polinomları, yüksek mertebeden iki parametreli Apostol-Euler polinomları, yüksek mertebeden iki parametreli Apostol-Genocchi polinomları ve Dikson polinomlarını içeren birçok bağıntılar ve formüller elde edilmiştir. Son olarak, bu polinomların bazı özel değerleri ve trigonometrik fonksiyonlarla uygulamaları sunulmuştur.

Anahtar Kelimeler

• Hermite tipli polinomlar
• Apostol tipli sayılar ve polinomlar
• parametrik tipli polinomlar
• özel sayılar ve polinomlar
• üreteç fonksiyonları

Kaynakça

1. Bretti, G. and Ricci, P. E., Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese Journal of Mathematics, 8, 3, 415-428, (2004).
2. Cesarano, C., Hermite polynomials and some generalizations on the heat equations, International Journal of Systems Applications, Engineering & Development, 8, 193-197, (2014).
3. Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G. and Torre, A., Theory of generalized Hermite polynomials, Computers & Mathematics with Applications, 28, 4, 71-83, (1994).
4. Dattoli, G., Lorenzutta, S., Maino, G., Torre, A. and Cesarano, C., Generalized Hermite polynomials and supergaussian forms, Journal of Mathematical Analysis and Applications, 203, 597-609, (1996).
5. Dere, R. and Simsek, Y., Hermite base Bernoulli type polynomials on the umbral algebra, Russian Journal of Mathematical Physics, 22, 1, 1-5, (2015).
6. Gould, W. and Hopper, A. T., Operational formulas connected with two generalizations of Hermite polynomials, Duke Mathematical Journal, 29, 51-62, (1962).
7. Hwang, K.-W. and Ryoo, C. S., Differential equations associated with two variable degenerate Hermite polynomials, Mathematics, 8, 228, (2020).
8. Kilar, N. and Simsek, Y., Computational formulas and identities for new classes of Hermite-based Milne-Thomson type polynomials: Analysis of generating functions with Euler's formula, Mathematical Methods in the Applied Sciences, 44, 6731-6762, (2021).

9. Kilar, N., Generating functions of Hermite type Milne-Thomson polynomials and their applications in computational sciences, PhD Thesis, Akdeniz University, Institute of Natural and Applied Sciences, Antalya, (2021).
10. Kurt, B., Explicit relations for the modified degenerate Apostol-type polynomials, Journal of Balıkesir University Institute of Science and Technology, 20, 2, 401-41, (2018).
11. Simsek, Y., Explicit formulas for p-adic integrals: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus Journal of Pure and Applied Mathematics, 1, 1, 1-76, (2019).
12. Srivastava, H. M. and Choi, J., Zeta and q-Zeta functions and associated series and integrals, Elsevier Science Publishers, Amsterdam, London and New York, (2012).
13. Srivastava, H. M. and Manocha, H. L., A treatise on generating functions, John Wiley and Sons/Ellis Horwood, New York/Chichester, (1984).
14. Srivastava, H. M., Masjed-Jamei, M. and Beyki, M. R., Some new generalizations and applications of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Rocky Mountain Journal of Mathematics, 49, 2, 681-697, (2019).
15. Dickson, L. E., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group I, II, Annals of Mathematics, 11, 1/6, 65-120, 161-183, (1897).
16. Rassias, Th. M., Srivastava, H. M. and Yanushauskas, A., Topics in polynomials of one and several variables and their applications, Word Scientific Publishing, Singapore, (1991).
17. Ryoo, C. S., Differential equations arising from the 3-variable Hermite polynomials and computation of their zeros, in Differential Equations-Theory and Current Research, Intech Open, 81-98, (2018).
18. Masjed-Jamei, M. and Koepf, W., Symbolic computation of some power trigonometric series, Journal of Symbolic Computation, 80, 2, 273-284, (2017).
19. Kilar, N. and Simsek, Y., Relations on Bernoulli and Euler polynomials related to trigonometric functions, Advanced Studies in Contemporary Mathematics (Kyungshang), 29, 2, 191-198, (2019).
20. Kilar, N. and Simsek Y., A special approach to derive new formulas for some special numbers and polynomials, Turkish Journal of Mathematics, 44, 2217-2240, (2020).
21. Kilar, N. and Simsek, Y., Identities and relations for Hermite-based Milne-Thomson polynomials associated with Fibonacci and Chebyshev polynomials, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, RACSAM, 115, 28, 1-20, (2021).
22. Kilar, N. and Simsek, Y., New computational formulas for special numbers and polynomials derived from applying trigonometric functions to generating functions, Milan Journal of Mathematics, 89, 217-239, (2021).
23. Rainville, E. D., Special functions, The Macmillan Company, New York, (1960).
24. Srivastava, H. M., Özarslan, M. A. and Kaanoğlu, C., Some families of generating functions for a certain class of three-variable polynomials, Integral Transforms and Special Functions, 21, 12, 885-896, (2010).